Ruxolitinib in Treating Participants With Chronic Myeloid Leukemia With Minimal Residual Disease While on Therapy With Tyrosine Kinase Inhibitors

Inclusion Criteria:

* Patients with Philadelphia chromosome (Ph)-positive or BCR/ABL-positive CML (as determined by cytogenetics, FISH, or PCR).
* Patients must be on continuous TKI therapy for management of their CML. Any commercially available and FDA- approved TKI can be used, i.e., imatinib mesylate (IM), nilotinib (NIL) or dasatinib (DAS). Patients may be receiving TKI at entry in the frontline or salvage setting, including patients currently on imatinib after alpha-interferon failure or on dasatinib or nilotinib after failure to prior therapy including imatinib.
* Patients must have received the current TKI for at least 18 months and not have increased their dose in the last 6 months.
* For the phase I portion of the study, patients may be included without a CCyR provided they remain in chronic or accelerated phase CML and have at least a complete hematologic response (CHR). For the Phase II portion of the study patients must be in complete cytogenetic remission (CCyR), regardless of the stage of disease they had at the time they started therapy with TKI.
* Patients must have detectable BCR-ABL transcript levels meeting at least 1 of the following criteria: Patient has never achieved a major molecular response (MMR, as defined by a BCR-ABL/ABL =\< 0.1% in the international scale (currently equivalent to 0.28 in the MD Anderson Cancer Center \[MDACC\] molecular diagnostic laboratory), and transcript levels have shown in at least 2 consecutive measures separated by at least 1 month to have increased by any value; or achieved a major molecular response which has been lost, with an interim increase in transcript levels by at least one-log, confirmed in two consecutive analyses separated by at least 1 month; or patient has received therapy for at least 2 years \& lacks a sustained major molecular response; or patient has received therapy for at least 5 years and lacks a sustained complete molecular response (CMR, defined as transcript levels still detectable in the MDACC molecular diagnostic laboratory).
* Patients must not have had a known interruption of TKI therapy of greater than 21 consecutive days or for a total of 6 weeks in the 6 months prior to enrollment.
* Patients must be able to understand and sign an informed consent indicating that they are aware of the investigational nature of this study in keeping with the institutional policies.
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2.
* Bilirubin \< 2 x upper limit of normal (ULN) (unless associated with Gilbert's syndrome).
* Alanine aminotransferase (ALT) or aspartate aminotransferase (AST) \< 2.5 x ULN.
* Absolute neutrophil count (ANC) \>= 1 x 10(9)/L.
* Platelets \>= 100 x 10(9)/L.
* Serum creatinine \< 1.5 mg/dL or creatinine clearance greater or equal than 60 cc/min as defined by the Cockcroft-Gault equation.
* Women of childbearing potential should be advised to avoid becoming pregnant while on therapy with ruxolitinib and for 30 days after the last dose and practice effective methods of contraception. Men should be advised not to father a child while receiving treatment with Ruxolitinib and for 30 days after the last dose. Effective methods of contraception for this study include barrier methods (e.g., condoms, diaphragm); spermicidal jelly or foam; oral, depo provera, or injectable hormonal contraceptives; intrauterine devices; tubal ligation; and abstinence.

Exclusion Criteria:

* For the phase I portion of the study, patients in blast phase. For the phase II portion of the study, patients in accelerated or blast phase.
* Patients receiving any other investigational agents.
* Patients who are pregnant or breast-feeding.
* Patients with clinically significant heart disease (New York Heart Association \[NYHA\] class III or IV).
* Patients with corrected QT (QTc) \> 480 msec.
* Patients taking a potent CYP3A4 inhibitor that cannot be changed to an alternate drug.
* Known or suspected hypersensitivity to ruxolitinib.
* Patients with advanced malignant hepatic tumors.
* Patients with known active hepatitis B or C, or human immunodeficiency virus (HIV) infection.
* Patients with other medical conditions or concomitant medications that in the opinion of the principal investigator may interfere with the therapeutic treatment.